AVR Freaks

Which indicates that your ISR is not being run or your port is not outputting anything because it is not configured correctly.

And just to prove the technique works...

/*******************************************************
Chip type               : ATmega1284P
Program type            : Application
AVR Core Clock frequency: 16.000000 MHz
Memory model            : Small
External RAM size       : 0
Data Stack size         : 4096
*******************************************************/

#include <mega1284p.h>
#include <delay.h>

// Timer 0 overflow interrupt service routine
interrupt [TIM0_OVF] void timer0_ovf_isr(void) {

	PORTA.0 = 0;
	TCNT0=0x25;
	delay_us(500);
    PORTA.0 = 1;
}

void main(void) {

    // Crystal Oscillator division factor: 1
    #pragma optsize-
    CLKPR=(1<<CLKPCE);
    CLKPR=(0<<CLKPCE) | (0<<CLKPS3) | (0<<CLKPS2) | (0<<CLKPS1) | (0<<CLKPS0);
    #ifdef _OPTIMIZE_SIZE_
    #pragma optsize+
    #endif

    // Input/Output Ports initialization
    // Port A initialization
    // Function: Bit7=In Bit6=In Bit5=In Bit4=In Bit3=In Bit2=In Bit1=In Bit0=Out
    DDRA=(0<<DDA7) | (0<<DDA6) | (0<<DDA5) | (0<<DDA4) | (0<<DDA3) | (0<<DDA2) | (0<<DDA1) | (1<<DDA0);
    // State: Bit7=T Bit6=T Bit5=T Bit4=T Bit3=T Bit2=T Bit1=T Bit0=0
    PORTA=(0<<PORTA7) | (0<<PORTA6) | (0<<PORTA5) | (0<<PORTA4) | (0<<PORTA3) | (0<<PORTA2) | (0<<PORTA1) | (0<<PORTA0);

    // Timer/Counter 0 initialization
    // Clock source: System Clock
    // Clock value: 62.500 kHz
    // Mode: Normal top=0xFF
    // OC0A output: Disconnected
    // OC0B output: Disconnected
    // Timer Period: 3.488 ms
    TCCR0A=(0<<COM0A1) | (0<<COM0A0) | (0<<COM0B1) | (0<<COM0B0) | (0<<WGM01) | (0<<WGM00);
    TCCR0B=(0<<WGM02) | (1<<CS02) | (0<<CS01) | (0<<CS00);
    TCNT0=0x25;
    OCR0A=0x00;
    OCR0B=0x00;

    // Timer/Counter 0 Interrupt(s) initialization
    TIMSK0=(0<<OCIE0B) | (0<<OCIE0A) | (1<<TOIE0);

    // Globally enable interrupts
    #asm("sei")

    while (1) {
		;
	}
}

Either you have failed to set up the trigger conditions for your scope correctly or the ISR simply is not being run.

Pulsing an IO to measure duration is a classic technique for determining the duration of a code sequence.

Try cutting the complexity out of your code and just do something in a main() loop. Maybe something like:

#include <stdlib.h>
#include <math.h>
#include <avr/io.h>
#include <util/delay.h>


volatile float f;

int main(void) {
    DDRC = (1 << 5);
    while(1) {
        PORTC |= (1 << 5);
        f = sin(rand());
        PORTC &= ~(1 << 5);
        _delay_ms(20);
    }
}

There should be a variable period high pulse on the IO line then 20ms low periods between. If you don't see this then your triggering is not working (or maybe you are even probing the wrong trigger pin?).

You do not have reasonable expectations of what an LED would show.

Any LED that flashes faster than about once every 100ms will appear steadily on, to your eye. It is called "persistence of vision". Instead, depending on the percentage of time the LED is on, it  will appear to have varying brightness, independent of the flash frequency.

Jim