Middle English thinken, a convergence of two Old English verbs from the same prehistoric source but with distinct forms and senses.
Thinken (1) "present the appearance of (something)" is from Old English þyncan, þincan. Thinken (2), "exercise the faculty of reason, cogitate" is from Old English þencan. Grammatically, þencan is the causative form of þyncan. The two converged in form in Middle English and the sense from þyncan "to seem" was absorbed or lost but is preserved in methinks "it seems to me."
The sense of "say to oneself mentally" (thinken (2)) was in Old English þencan "imagine, conceive in the mind; consider, meditate, remember; intend, wish, desire" (past tense þohte, past participle geþoht), probably originally "cause to appear to oneself," from Proto-Germanic *thankjanan (source also of Old Frisian thinka, Old Saxon thenkian, Old High German denchen, German denken, Old Norse þekkja, Gothic þagkjan).
Old English þyncan "to seem, to appear" (past tense þuhte, past participle geþuht) is the source of Middle English thinken (1). It is reconstructed to be from Proto-Germanic *thunkjan (source also of German dünken, däuchte).
Both are from PIE *tong- "to think, feel" (Watkins), which also is the root of thought and thank. Boutkan gives this no IE origin, rejects proposed cognates, and suggests a substrate source.
Thinken (1) in Middle English also could mean "seem erroneously or falsely" or "seem fitting or proper." It often was used impersonally, with an indirect object, as in methinks.
To think twice "hesitate, reconsider" is by 1898; to think on one's feet "adjust quickly to changing circumstances" is by 1935; to think so "be of that opinion" is by 1590s; to think (something) over "give continued thought to" is by 1847. To think up "invent, make up, compose" is from early 15c. Modern use might be 19c. I tink, representing dialectal or foreign pronunciation of "I think," is by 1767.