How many bits are needed for an LFSR to generate a specific number of sequences? I need to generate 64-bit data sequences 16 times pseudorandomly. How many bits should the linear-feedback shift register (LFSR) have in order to do so? 

That is, what is the number of bits for the initial seed? How can I calculate this number? Can we minimize this number? Should it be a fixed number of bits?
An $n$-bit LFSR with a primitive feedback polynomial will cycle through all $2^n-1$ possible non-zero states. You will need an LFSR to generate $16\times 64 = 1024$ bits. An 11-bit LFSR will cycle through $2^{11} - 1 = 2047$ states and output as many bits, so for your purposes, an 11-bit state is sufficient.

If the feedback polynomial is not primitive, then the period will be unpredictable but not maximal. An example primitive polynomial suitable for use in a maximal-length 11-bit LFSR is $x^{11}+x^9+1$.

It's extremely important to know that an LFSR is not cryptographically secure. An $n$-bit LFSR can be broken after only $n$ bits of keystream are known ($2n$ bits if the feedback polynomial is secret).