Are there Biblical examples of being adopted into an Israelite tribe?

Ruth as a model

The clearest example is Ruth. She was a Moabite and thus forbidden by law as a wife to Israelites.

No Ammonite or Moabite or any of their descendants may enter the assembly of the Lord, not even in the tenth generation. (Deuteronomy 23:3)

Ruth's mother-in-law was a member of the tribe of Judah living in Moab, and one of her sons married Ruth. When the son died, Naomi decided to return to her hometown and told Ruth to stay in Moab. But Ruth said:

Don’t urge me to leave you or to turn back from you. Where you go I will go, and where you stay I will stay. Your people will be my people and your God my God. Where you die I will die, and there I will be buried. May the Lord deal with me, be it ever so severely, if even death separates you and me. (Ruth 1:16-17)

To make a long story short, she eventually married Naomi's kinsman, Boaz, and became the great-grandmother of King David. She thus became a member of the tribe of Judah. It is presumed she adopted all Israelite customs.

The process of conversion

The Law itself does not specify the process of conversion but later traditions involved an immersion in a ritual bath, similar to baptism, as well as circumcision for and offering a sacrifice. The Jewish Encyclopedia explains:

According to rabbinical teachings, which dominated even during the existence of the Temple (Pes. viii. 8), Baptism, next to circumcision and sacrifice, was an absolutely necessary condition to be fulfilled by a proselyte to Judaism... Circumcision, however, was much more important, and, like baptism, was called a "seal".

We cannot say for certain how strictly these rules were enforced in the early days of Israel. Adoption into a tribe may have more informal in those days.

Mass conversions and gradual assimilation

In some cases whole clans were adopted into Israel. The story of the Gibeonites' mass conversion is one such case. The Gibeonites plead with Joshua:

“Your servants were clearly told how the Lord your God had commanded his servant Moses to give you the whole land and to wipe out all its inhabitants from before you. So we feared for our lives because of you... We are now in your hands. Do to us whatever seems good and right to you."

So Joshua saved them from the Israelites, and they did not kill them. That day he made the Gibeonites woodcutters and water carriers for the assembly, to provide for the needs of the altar of the Lord at the place the Lord would choose. And that is what they are to this day. (Josh 9:24-27)

The biblical conquest of Canaan left many native people remaining the land. A large number of these would have gradually converted to Judaism though assimilation. Some, like the Gibeonites were enslaved first. 1 Kgs 9:20-21 describes Solomon's incorporating the remaining Canaanite tribes through forced labor:

All the people that were left of the Amorites, Hittites, Perizzites, Hivites, and Jebusites who were not of the Israelite stock—those of their descendants who were still left in the land, whom the Israelites were unable to destroy completely—of these Solomon made a slave force, as is still the case.

Further reading:

The Jewish Encyclopedia's article titled Proselyte contains a good deal of useful information on this, including:

Ex. xii. 48 provides for the proselyte's partaking of the paschal lamb, referring to him as a "ger" that is "circumcised." Isa. xiv. 1 mentions converts as "strangers" who shall "cleave to the house of Jacob"... Deut. xxiii. 8 speaks of "one who enters into the assembly of Jacob," and (Deutero-) Isa. lvi. 3-6 enlarges on the attitude of those that joined themselves to Yhwh, "to minister to Him and love His name, to be His servant, keeping the Sabbath from profaning it, and laying hold on His covenant." "Nokri" (ξένος ="stranger") is another equivalent for "proselyte," meaning one who, like Ruth, seeks refuge under the wings of Yhwh (Ruth ii. 11-12; comp. Isa. ii. 2-4, xliv. 5; Jer. iii. 17, iv. 2, xii. 16; Zeph. iii. 9; I Kings viii. 41-43; Ruth i. 16).

See also the Wikipedia article: Conversion to Judaism